If $z_1 = 1+2i$ and $z_2 = 3+5i$ , then ${\mathop{\rm Re}\nolimits} \,\left( {\frac{{{{\overline Z }_2}{Z_1}}}{{{Z_2}}}} \right) = $
$\frac {-31}{17}$
$\frac {17}{22}$
$\frac {-17}{31}$
$\frac {22}{17}$
$arg\,(5 - \sqrt 3 i) = $
If $z$ is a complex number, then the minimum value of $|z| + |z - 1|$ is
Let $z$, $w \in C$ satisfy ${z^2} + \bar w = z$ and ${w^2} + \bar z = w$ then number of ordered pairs of complex numbers $(z, w)$ is equal to
If $|z_1| = 2 , |z_2| =3 , |z_3| = 4$ and $|2z_1 +3z_2 +4z_3| =9$ ,then value of $|8z_2z_3 +27z_3z_1 +64z_1z_2|$ is equal to:-
If $arg\,z < 0$ then $arg\,( - z) - arg\,(z)$ is equal to